inches and a half in compass; the assize of which is to be enquired of by justices. BILLETTING, in military affairs, is the quartering of soldiers in the houses of a town or village. BILLIARDS, an ingenious kind of game, played on a rectangular table, with little ivory balls, which are driven into hazards or holes, according to certain rules of the game. The table on which the game is played is generally about twelve feet long and six feet wide, or rather in the exact form of an oblong; it is covered with fine green cloth, and surrounded with cushions, to prevent the balls rolling off, and to make them rebound. There are six holes, nets, or pockets: these are nixed at the four corners, and in the middle, opposite to each other, to receive the balls, which, when put into these holes or pockets, are called hazards. The making of a hazard, that is, putting the adversary's ball in, at the usual game, reckons for two in favour of the player. The game is played with sticks called maces, or with cues; the first consists of a long straight stick, with a head at the end, and are the most powerful instruments of the two: the cue is a thick stick, diminishing gradually to a point of about half an inch diameter; this instrument is played over the left hand, and supported by the fore-hinger and thumb. It is the only instrument in vogue abroad, and is played with amazing address by the Italians and some of the Dutch; but in England the mace is the prevailing instrument, which the foreigners hold in contempt, as it requires not near so much address to play the game with, as when the cue is made use of; but the mace is preferred for its peculiar advantage which some professed players have artfully introduced, under the name of trailing, that is, following the ball with the mace to such a convenient distance from the other ball as to make it an easy hazard. The degrees of trailing are various, and undergo different denomina. tions amongst the connoisseurs at this game; viz. the shove, the sweep, the long stroke, the trail, and the dead trail, or turn up, all which secure an advantage to a good player, according to their various gradations: even the butt end of the cue becomes very powerful, when it is made use of by a good trailer. Rules generally observed at the common or usual game:-1. For the lead, the balls must be put at one end, and the player must strike them against the farthermost cushion, in order to see which will be nearest the cushion that is next to them. But 2. The nearest to the cushion is to lead, 19. Different kinds of games played at billiards.- Besides the common winning game, which is twelve up, there are several other kinds of game, viz. the losing game, the winning and losing, choice of balls, bricole, carambole, Russian carambole, the barhole, the one-hole, the fourgame, and hazards: but on these it is not necessary to enlarge. BINARY arithmetic, that wherein unity, or 1 and 0, are only used. This was the invention of Mr. Liebnitz, who shows it to be very expeditious in discovering the properties of numbers, and in constructing tables; and Mr. Dangecourt, in the "History of the Royal Academy of Scien ces," gives a specimen of it concerning arithmetical progressionals; where he shews that, because in binary arithmetic only two characters are used, therefore the laws of progression may be more easily discovered by it than by common arithmetic. All the characters used in binary arithmetic are 0 and 1, and the cypher multiplies every thing by 2, as in the common arithmetic by 10. Thus, 1 is one; 10, two; 11, three; 100, four, 101 five; 110, six; 111, seven; 1000, eight; 1001, nine; 1010, ten, which is built on the same principles with common arithmetic. The author, however, does not recommend this method for common use, because of the great number of figures required to express a number; and adds, that if the common progression were from 12 to 12, or from 16 to 16, it would be still more expeditious. decrease gradually by 'the same differences, viz. unit, and that in the last terms it is never found. The powers of bare in the contrary order; it is never found in the first term, but its exponent in the second term is unit; in the third term its exponent is 2; and thus its exponent increases, till in the last term it becomes equal to the exponent of the power required. and at the same time those of b increase, As the exponents of a thus decrease, the sum of their exponents is always the same, and is equal to the exponent of the power required. Thus in the sixth power of a+b, viz. a6+6a5 b+15 a+ b2+20 a3 b3 +15 a2 b++6ab6+ b, the exponents of a decrease in this order 6, 5, 4, 3, 2, 1, 0; and those of b increase in the contrary order 0, 1, 2, 3, 4, 5, 6. And the sum of their exponents in any term is always 6. raised to any power m, the terms without In general, therefore, if a+b is to be their coefficients will be am, ɑm—'b, am— 1b3, am—3b3, um—4b4, am—5b5, &c. continued till the exponent of b become equal to m. X It follows therefore by these rules, that a+6mɑm+mam BIND weed. See CONVOLVULUS. BINOMIAL, in algebra, a root consisting of two members, connected by the sign+or-. Thus ab and 8-3 are binomials; consisting of the sums and am 2f2+m × differences of these quantities. The powers of any binomial are found by a continual multiplication of it by itself. For example the cube or third power of a+b, will be found by multiplication to be a3+3 a2b+3ab+b3; and if the powers of a-b are required, they will be found the same as the preceding, only the terms in which the exponent of b is an odd number will be found negative. Thus the cube of a-b will be found to be a33a-b-3ab-b3, where the second and fourth terms are negative, the exponent of b being an odd number in these terms. In general the terms of any power of a-b are positive and negative by turns. It is to be observed that in the first term of any power of a+b, the quantity a has the exponent of the power required, that in the following terms the exponents of a m-1 I b+m+ m2 + m-1 m-2 X am m-2 m-3 b3+m x X X 2 3 4 X am-4b4+, &c. which is the binomial or general theorem for raising a quantity consisting of two terms to any power m. The same general theorem will also serve for the evolution of binomials, because to extract any root of a given quantity is the same thing as to raise that quantity to a power, whose exponent is a fraction that has its denominator equal to the number that expresses what kind of root is to be extracted. Thus, to extract the square root of a + b, is to raise a+b to a power whose exponent is . Now a+bm being found as above; suppos. ing 1 m, you will find a+b == a+ - } b + 1 × − 1 × a− 32 b2 + 1 × a &c. To investigate this theorem, suppose n quantities, + a, x + b, x + c, &c. multiplied together; it is manifest that the first term of the product will be an, and that 1, xn-2, &c. the other powers of x, will all be found in the remaining terms, with different combinations of a, b, c, d, &c. Let x+b.x + c.x+d. &c.xn- B = a.b+c+d+&c. +c.d+&c. In the same manner, Ca.b.c.+d+ &c. +a.c. d+ &c. +b.c.d+ &c. and so on; that is, A is the sum of the quantities a, b, c, &c. B is the sum of the products of every two; C is the sum of the products of every three, &c. &c. If the index of the power to which a binomial is to be raised be a whole positive number, the series will terminate, be. n-1 n- -2 cause the co-efficient n. 3 3 &c. will become nothing when it is continued to n+1 factors. In all other cases the number of terms will be indefinite. When the index is a whole positive number, the co-efficients of the terms ta ken backward, from the end of the series, are respectively equal to the co-efficients of the corresponding terms taken forward from the beginning. Thus, in the first example, where a+r Let a = b = c = d = &c. then A, is raised to the 8th power, the co-efficients or a+b+c+d+ &c. =na;= abare, 1, 8, 28, 56, 70, 56, 28, 8, 1. ac B+be+ &c. = a2 X the number of In general, the co-efficient of the n+1th combinations of a, b, c, d, &c. taken two n.n- -1.n—2... 3.2.1 n-1 term is 1.2.3...... n-2. n-1.n n.n-1.n The co-efficient of the nth term is 1.2.3.... of the n-1th term, and two,n. 2 in the same manner it appears that C = n. 2 -a3, &c. 3 And x+a.x+b.x+c. &c. to n factors =x+al; therefore x + a n = xn+n n-1 -a2x22+n. N- -1 2 -2....3.2. n-2.n-1 = n; 1. The sum of the co-efficients 1+n+n. For if x=a=1, then x+an = 1+1 =2′′=1+n+n.”—1+ &c. n Since x+an xn + na xn−1+n. -1 N- a2 xn+ &c. 2 And x-an-x”—n a xn—}, a2 xn -&c. 91 2 -1 By addition, x+an+x—a}" = 2.xn cords; and it is from private life, from familiar, domestic, and apparently trivial occurrences, that we often derive the most accurate knowledge of the real character. The subjects of Biography are not only the lives of public or private persons, who have been eminent and beneficial to the world, but those also of persons notorious for their vice and profligacy, which may serve, when justly characterised, as warnings to others, by exhibiting the fatal consequences, which, sooner or later, generally follow licentious practices. As for those who have exposed their lives, or devoted their time and talents, for the service of their fellowcreatures, it is but a debt of gratitude to +n. perpetuate their memories, by making posterity acquainted with their merits and usefulness. In the lives of public persons, their public characters are principally, but not solely, to be regarded; the world is interested in the minutest actions of great men, and their examples, both as public and private characters, may be made subservient to the well being and prosperity of society. n- 1 a2 xn+ &c. 2 -а3 xn' 3+ &c. n -1 2 The trinomial a+b+c may be raised to any power by considering two terms as one factor, and proceeding as before. Thus, a+b+cn=an+n. b+c. an ̄1 n- -1 +n. b+c2. an2 + &c. and 2 the powers of b+c may be determined by the binomial theorem. BIOGRAPHY, a very entertaining and instructive species of history, containing the life of some remarkable person or persons. : Lord Bacon regrets, that the lives of eminent men are not more frequently written for, adds he, though kings, princes, and great personages, be few; yet there are many other excellent men, who deserve better than vague reports and barren elogies. Biography, or the art of describing and writing lives, is a branch or species of history, in many respects as useful and important as that of history itself; inasmuch as it represents great men more distinctly, unencumbered with associates; and descending into the detail of their actions and characters, their virtues and failings, we obtain a more particular, and, of course, a more interesting acquaintance with individuals, than general his tory allows A writer of lives may, and ought, to descend to minute circumstances and familiar incidents. He is expected to give the private, as well as the public, life of those whose actions he re It has been a matter of dispute among the learned, whether any one ought to write his own history. There are instances, both ancient and modern, that may be adduced as precedents for the practice: and the reason assigned for it is, that no man can be so much the master of the subject as the person himself: cult task for any one to write an impartial but, on the other hand, it is a very diffihistory of his own actions. Plutarch mentions two cases, in which it is allowable for a man to commend himself, and to be the publisher of his own merits; which are, when the doing of it may be of considerable advantage either to himself or to others. Notwithstanding this high authority, the former case is unquestionably liable to great objections, because a man is to be the judge in his own cause, and therefore very liable to exceed the limits of truth, when his own interests are concerned, and when he wishes to render himself conspicuous for virtue or talents The ancients, however, had a peculiar method of diverting the reader's attention from themselves, when they had occasion to record their own actions, and of thus rendering what they said less invidious, which was, by speaking of themselves in the third person. Among the moderns a practice has been introduced, which cannot be too strongly reprobated, though sanctioned by men of great talent, integrity, and real worth; namely, of making the memoirs of themselves the vehicle of abuse of their contemporaries, every one of whom would, no doubt, be able to give a very different and perhaps plausible reason, for the several actions which the biographer has undertaken to scrutinize and condemn. It con Dr. Priestley has constructed and published a “Biographical Chart,” of which our plate is given as a specimen. This chart represents the interval of time between the year 1200 before the Christian æra, and 1800 after Christ, divided by an equal scale into centuries. tains about 2000 names of persons, the most distinguished in the annals of fame, the length of whose lives is represented by lines drawn in proportion to their real duration, and terminated in such a manner as to correspond to the dates of their births and deaths. These names are distinguished into several classes by parallel lines running the whole length of the chart, the contents of each division being expressed at the end of it. The chronology is noted in the margin, on the upper side, by the year before and after Christ, and on the lower by the same æra, and also by the succession of such kings as were most distinguished in the whole period. See Plate BIOGRA PHY. For a more full account we refer to Dr. Priestley's description, which accompanies the chart; from which we shall make a short extract, that cannot fail to entertain the reader. "Laborious and tedious as the compilation of this work has been (vastly more so than my first conceptions represented it to me,) a variety of views were continually opening upon me during the execution of it, which made me less attentive to the labour. As these views agreeably amuse the mind, and may, in some measure, be enjoyed by a person who only peruses the chart, without the labour of compilation, I shall mention a few of them in this place. "It is a peculiar kind of pleasure we receive, from such a view as this chart exhibits of a great man, such as Sir Isaac Newton, seated, as it were, in the circle of his friends and illustrious contemporaries. We see at once with whom he was capable of holding conversation, and in a manner (from the distinct view of their respective ages) upon what terms they might converse. And though it be melancholy, it is not unpleasing, to observe the order in which we here see illustrious persons go off the stage, and to imagine to ourselves the reflections they VOL. H. might make upon the successive departure of their acquaintance or rivals. "We likewise see in some measure, by the names which precede any person, what advantages he enjoyed from the labours and discoveries of others; and, by those which follow him, of what use his labours were to bis successors. "By the several void spaces between such groups of great men, we have a clear idea of the great revolutions of all kinds of science, from the very origin of it; so that the thin and void places in the chart are, in fact, no less instructive than the most crowded, in giving us an idea of the great interruptions of science, and the intervals at which it hath flourished. The state of all the divisions appropriated to men of learning is, for many centuries before the revival of letters in this western part of the world, exactly expressed by this following line of Virgil: Apparent rari nantes in gurgite vasto. But we see no void spaces in the division of statesmen, heroes, and politicians. The world hath never wanted competitors for empire and power, and least of all in those periods in which the sciences and the arts have been the most neglected. "But the noblest prospect of this nature is suggested by a view of the crouds of names, in the divisions appropriated to the arts and sciences in the two last centuries. Here all the classes of renown, and, I may add, of merit, are full; and a hundred times as many might have been admitted, of equal attainments in knowledge with their predecessors. This prospect gives us a kind of security for the continual propagation and extension of knowledge; and that, for the future, no more great chasms of men, really eminent for knowledge, will ever disfigure that part of the chart of their lives which I cannot draw, or ever see drawn. What a figure must science make, advancing as it now does, at the end of as many centuries as have elapsed since the Augustan age!" BIPED, in zoology, an animal furnished with only two legs. Men and birds are bipeds. Apes occasionally walk on their hind legs, and seem to be of this tribe; but that is not a natural position for them, and they rest upon all their legs, like other quadrupeds. The jerboas are also of the latter description, jumping and leaping on their hind legs, but resting on their fore legs likewise. BIQUADRATIC power, in algebra, the fourth power or squared square of a number, as 16 is the biquadratic power of Bb |